A main concept of algebraic geometry, having a roughly equivalent role to that of a manifold in differential geometry. An affine variety in Fⁿ, where F is a field, is a subset defined by polynomial equations in x₁,…,x_n. So conics and quadrics are examples of varieties. Projective varieties are similarly defined by homogeneous polynomial equations in the coordinates. Commonly, the field will be assumed to be algebraically closed (such as ℂ), as this results in a richer theory, such as Bézout’s theorem. It is also possible to define varieties abstractly, making no reference to an ambient space.